A394018 - OEIS

%I #12 Mar 13 2026 17:22:30

%S 1,0,0,0,1,0,0,1,1,0,0,1,2,0,1,1,2,0,1,3,2,1,1,3,3,1,3,3,5,1,3,5,5,3,

%T 3,7,6,3,6,8,9,3,8,10,9,6,9,14,10,8,12,15,15,9,16,17,18,13,18,23,20,

%U 17,21,27,26,19,28,32,31,23,33,40,34,30,39,47,41,36

%N Expansion of g.f.: Product_{k>=1} 1 / (1 - x^(k^2 + 3)).

%C a(n) is the number of partitions of n into parts of the form k^2+3 (for k>=1).

%H Vaclav Kotesovec, <a href="/A394018/b394018.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) ~ zeta(3/2)^(2/3) * exp(3*Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3)) / (2^(7/3) * Pi^(1/6) * sinh(Pi*sqrt(3)) * n^(7/6)) * (1 - (34 + 27*Pi * zeta(1/2) * zeta(3/2)) / (9*2^(5/3) * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3))).

%t nmax = 150; CoefficientList[Series[1/Product[1 - x^(k^2 + 3), {k, 1, Floor[Sqrt[nmax]+1]}], {x, 0, nmax}], x]

%Y Cf. A001156, A393990, A394017, A394019, A394020.

%K nonn

%O 0,13

%A _Vaclav Kotesovec_, Mar 06 2026